Rotating-convection growth-rate polynomial

For stress-free impermeable fixed-temperature plates, thermal-time growth rate $s$, horizontal wavenumber $k$ and vertical index $n$, put $a=k^2+n^2{\pi }^2$ and $P=\mathrm{Pr}$. Vertical velocity, temperature and vertical vorticity amplitudes satisfy $a(s/P+a)W+\sqrt{\mathrm{Ta}}n\pi Z=R{k}^2\Theta$, $(s/P+a)Z=\sqrt{\mathrm{Ta}}n\pi W$, $(s+a)\Theta =W$. Taking their determinant givesThis is a cubic and remains valid without dividing by a potentially zero factor. Setting $s=0$ or $s=i\omega$ determines the stationary or admissible oscillatory neutral curves.

For $s=i\omega$, real-imaginary separation of the rotating-convection cubic givesThe curve is admissible only if ${\omega }^2>0$, so $P<1$ and sufficient rotation are necessary. A formal minimum with nonpositive angular frequency squared is not a Hopf bifurcation. At zero angular frequency this curve meets the stationary curve for that fixed wavenumber in a double-zero limit.

Substitution of zero growth rate into the rotating-convection cubic gives the displayed Rayleigh number. For the first vertical mode, setting $t=k^2$, $m={\pi }^2$, differentiating with respect to $t$ gives $(t+m{)}^2(2t-m)=\mathrm{Ta}m$. At rapid rotation, $k_c\sim (\mathrm{Ta}{\pi }^2/2{)}^{1/6}$ and $R_c\sim 3(\mathrm{Ta}{\pi }^2/2{)}^{2/3}$. Discrete allowed wavenumbers or different plate conditions require a different minimization.